No, it cannot. Assuming there is a Pokemon in your party that has Pokerus, every time a battle concludes there is a chance it will spread to the occupied Pokemon slots next to them in your party. Since you can't conclude a battle in sleep mode, it's generally impossible.
In case you thought putting an infected Pokemon in the PC Box would spread to Pokemon over the PC Box:
>If you put them in a PC box it actually paused the decay rate of the virus allowing you to preserve it for further spread. The virus only lasts 1-4 days, after which the Pokemon is immune to it and cannot spread it, but still gets its benefits. So always have several Pokemon in the PC boxes that are infected for further spread.
So, you can't put it in the PC Box if you want it to spread. Only in your party.
Hope I helped. :)